Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal 1-bit Error Exponent for 2-hop Relaying with Binary-Input Channels

Yan Hao Ling, Jonathan Scarlett

TL;DR

The paper addresses the problem of relaying a single bit over a tandem of binary-input DMCs to maximize the error exponent of the decoding error probability. Building on the TeachLearn framework, it derives a tight converse that matches the existing achievability bound, proving that the 2-hop exponent E(P,Q) equals the Chernoff-divergence based min-max expression across the two channels. The authors introduce a refined analytic framework using tilted distributions, tensorization of Chernoff bounds, and a prefix-free set to structure a three-case argument, ultimately showing that -log(p_e0+p_e1) scales as nE^*+o(n) with E^* = max_s min{d_C(P_0,P_1,s), d_C(Q_0,Q_1,s)}. This result closes gaps from the prior work and provides a complete characterization of the optimal 2-hop exponent for binary-input channels, with implications for relay strategies in binary-network settings. The work also highlights limitations to binary inputs and outlines directions for extending the analysis to larger input alphabets as future research.

Abstract

In this paper, we study the problem of relaying a single bit over a tandem of binary-input channels, with the goal of attaining the highest possible error exponent in the exponentially decaying error probability. Our previous work gave an exact characterization of the best possible error exponent in various special cases, including when the two channels are identical, but the general case was left as an open problem. We resolve this open problem by deriving a new converse bound that matches our existing achievability bound.

Optimal 1-bit Error Exponent for 2-hop Relaying with Binary-Input Channels

TL;DR

The paper addresses the problem of relaying a single bit over a tandem of binary-input DMCs to maximize the error exponent of the decoding error probability. Building on the TeachLearn framework, it derives a tight converse that matches the existing achievability bound, proving that the 2-hop exponent E(P,Q) equals the Chernoff-divergence based min-max expression across the two channels. The authors introduce a refined analytic framework using tilted distributions, tensorization of Chernoff bounds, and a prefix-free set to structure a three-case argument, ultimately showing that -log(p_e0+p_e1) scales as nE^*+o(n) with E^* = max_s min{d_C(P_0,P_1,s), d_C(Q_0,Q_1,s)}. This result closes gaps from the prior work and provides a complete characterization of the optimal 2-hop exponent for binary-input channels, with implications for relay strategies in binary-network settings. The work also highlights limitations to binary inputs and outlines directions for extending the analysis to larger input alphabets as future research.

Abstract

In this paper, we study the problem of relaying a single bit over a tandem of binary-input channels, with the goal of attaining the highest possible error exponent in the exponentially decaying error probability. Our previous work gave an exact characterization of the best possible error exponent in various special cases, including when the two channels are identical, but the general case was left as an open problem. We resolve this open problem by deriving a new converse bound that matches our existing achievability bound.
Paper Structure (12 sections, 24 theorems, 129 equations, 2 figures)

This paper contains 12 sections, 24 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Statement of Main Result
  4. Other Related Work
  5. Preliminaries
  6. Further Definitions and Notation
  7. Chernoff Divergence and its Properties
  8. Error Bounds for 1-Hop Communication
  9. Intuitive Overview of the Proof
  10. Simplifying Conditions Towards Proving Theorem \ref{['thm:binary_optimal_exact']}
  11. Completion of the Proof of Theorem \ref{['thm:binary_optimal_exact']}
  12. Conclusion

Key Result

Theorem 1

(teachlearn) Let $P, Q$ be binary input DMCs, and let $P_0, P_1, Q_0, Q_1$ represent the conditional output distributions of $P$ and $Q$ for inputs $0,1$ respectively. ThenHaving two inner "max" operations in eq:achievability is redundant, because if we replace one of them (but not the other) by the where $d_{\rm C}(P_0.P_1,s) = -\log \sum_{x \in \mathcal{X}} P_0(x)^{1-s}P_1(x)^{s}$ for $s \in (0,

Figures (2)

  • Figure 1: Illustration of our problem setup.
  • Figure 2: Examples of skewed, balanced, and neutral. In this case, we are using $s^*=0.4$ (and therefore $1-s^*=0.6$). The solid circle represents the global maximum.

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • Lemma 9
  • ...and 34 more